Group #21456333 — subgroups

Subgroups — #21456333

Order 7808.

356 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 173 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16840319	C₂	1	2	yes	maximal
2	#16840167	C₂	1	2	yes	maximal, complemented
3	#16840164	C₂	1	4	yes	maximal, complemented
4	#16840160	C₂	1	2	yes	maximal, complemented
5	#16839800	C₂	1	4	yes	maximal
6	C₂×C₄×C₈×C₆₁	C₂	1	1	yes	maximal, characteristic
7	#14519777	C₂²	1	4	yes	—
8	#14519769	C₂²	1	4	yes	—
9	#14519756	—	2	4	no	—
10	C₄×Dic₁₂₂	—	2	4	no	—
11	#14519736	C₂²	1	4	yes	—
12	#14519711	C₂²	1	8	yes	—
13	#14519709	C₂²	1	8	yes	—
14	C₂²×C₈×C₆₁	C₂²	1	2	yes	—
15	C₂×C₄²×C₆₁	C₂²	1	1	yes	characteristic
16	C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
17	C₄×C₈×C₆₁	C₂²	1	2	yes	—
18	C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
19	C₂²×C₄×C₆₁	D₄	1	2	yes	—
20	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
21	C₂×Dic₁₂₂	—	2	4	no	—
22	C₂×C₈×C₆₁	Q₈	1	8	yes	complemented
23	C₂×C₈×C₆₁	C₂³	1	2	yes	—
24	C₂×C₈×C₆₁	C₂³	1	2	yes	—
25	C₄²×C₆₁	D₄	1	2	yes	—
26	C₄²×C₆₁	C₂³	1	1	yes	characteristic
27	C₄²×C₆₁	C₂³	1	1	yes	characteristic
28	C₂²⋊Dic₆₁	—	2	4	no	—
29	C₄⋊Dic₆₁	—	2	4	no	—
30	C₄⋊Dic₆₁	C₂³	1	8	yes	—
31	#11758196	—	2	8	no	—
32	C₄×Dic₆₁	—	2	4	no	—
33	C₂³×C₆₁	C₂×D₄	1	1	yes	characteristic
34	C₂×C₄×C₆₁	—	2	4	no	—
35	C₂×C₄×C₆₁	C₂×D₄	1	2	yes	—
36	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₆₁	C₂×D₄	1	2	yes	—
38	C₂×C₄×C₆₁	C₂×Q₈	1	4	yes	—
39	C₂×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
40	C₂×Dic₆₁	—	2	8	no	—
41	Dic₁₂₂	—	4	8	no	—
42	C₈×C₆₁	C₂×Q₈	1	4	yes	—
43	C₈×C₆₁	C₂×Q₈	1	4	yes	—
44	C₂²×C₆₁	—	2	2	no	—
45	C₂²×C₆₁	C₄⋊Q₈	1	2	yes	—
46	C₂²×C₆₁	C₂²×D₄	1	1	yes	characteristic
47	C₄×C₆₁	—	2	2	no	—
48	C₄×C₆₁	C₄⋊Q₈	1	2	yes	—
49	C₄×C₆₁	#135	1	4	yes	—
50	C₄×C₆₁	C₂²×Q₈	1	2	yes	—
51	Dic₆₁	—	4	8	no	—
52	#3160	—	61	1	no	maximal, Sylow
53	C₂×C₆₁	—	2	2	no	—
54	C₂×C₆₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
55	C₂×C₆₁	C₂×#135	1	2	yes	—
56	#533	—	61	2	no	—
57	#500	—	61	2	no	—
58	#499	—	61	4	no	—
59	#498	—	61	2	no	—
60	#427	—	61	4	no	—
61	C₂×C₄×C₈	D₆₁	1	1	yes	characteristic
62	C₆₁	#3160	1	1	yes	characteristic, complemented, Sylow
63	C₂²×C₈	C₂×D₆₁	1	2	yes	—
64	C₄⋊Q₈	—	61	4	no	—
65	#125	—	61	4	no	—
66	C₂²⋊Q₈	—	122	4	no	—
67	C₄×Q₈	—	122	4	no	—
68	#117	—	61	4	no	—
69	C₂×C₄²	C₂×D₆₁	1	1	yes	characteristic
70	#107	—	61	8	no	—
71	#106	—	61	8	no	—
72	C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
73	C₄×C₈	C₂×D₆₁	1	2	yes	—
74	C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
75	C₂×Q₈	—	122	4	no	—
76	C₂²×C₄	D₂₄₄	1	2	yes	—
77	C₂²×C₄	C₂²×D₆₁	1	1	yes	characteristic
78	C₂×C₈	Dic₁₂₂	1	8	yes	complemented
79	C₂×C₈	C₂²×D₆₁	1	2	yes	—
80	C₂×C₈	C₂²×D₆₁	1	2	yes	—
81	C₄⋊C₄	—	61	8	no	—
82	C₄⋊C₄	—	122	8	no	—
83	C₄⋊C₄	—	122	4	no	—
84	C₂²⋊C₄	—	122	4	no	—
85	C₄²	—	122	4	no	—
86	C₄²	D₂₄₄	1	2	yes	—
87	C₄²	C₂²×D₆₁	1	1	yes	characteristic
88	C₄²	C₂²×D₆₁	1	1	yes	characteristic
89	C₂³	C₂×D₂₄₄	1	1	yes	characteristic
90	Q₈	—	244	8	no	—
91	C₂×C₄	—	2	4	no	—
92	C₂×C₄	—	122	8	no	—
93	C₂×C₄	C₂×Dic₁₂₂	1	4	yes	—
94	C₂×C₄	C₂×D₂₄₄	1	2	yes	—
95	C₂×C₄	C₂×D₂₄₄	1	2	yes	—
96	C₂×C₄	C₂×D₂₄₄	1	1	yes	characteristic
97	C₂×C₄	C₂³×D₆₁	1	1	yes	characteristic
98	C₈	C₂×Dic₁₂₂	1	4	yes	—
99	C₈	C₂×Dic₁₂₂	1	4	yes	—
100	C₂²	—	2	2	no	—
101	C₂²	#14519777	1	2	yes	—
102	C₂²	C₂²×D₂₄₄	1	1	yes	characteristic
103	C₄	—	2	2	no	—
104	C₄	—	244	8	no	—
105	C₄	#14519777	1	2	yes	—
106	C₄	#14519799	1	4	yes	—
107	C₄	C₂²×Dic₁₂₂	1	2	yes	—
108	C₂	—	2	2	no	—
109	C₂	C₂×#14519777	1	1	yes	characteristic
110	C₂	C₂×#14519799	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#16840319

C₂

yes

maximal

#16840167

C₂

yes

maximal, complemented

#16840164

C₂

yes

maximal, complemented

#16840160

C₂

yes

maximal, complemented

#16839800

C₂

yes

maximal

C₂×C₄×C₈×C₆₁

C₂

yes

maximal, characteristic

#14519777

C₂²

yes

—

#14519769

C₂²

yes

—

#14519756

—

C₄×Dic₁₂₂

—

#14519736

C₂²

yes

—

#14519711

C₂²

yes

—

#14519709

C₂²

yes

—

C₂²×C₈×C₆₁

C₂²

yes

—

C₂×C₄²×C₆₁

C₂²

yes

characteristic

C₄×C₈×C₆₁

C₂²

yes

characteristic

C₄×C₈×C₆₁

C₂²

yes

—

C₄×C₈×C₆₁

C₂²

yes

characteristic

C₂²×C₄×C₆₁

D₄

yes

—

C₂²×C₄×C₆₁

C₂³

yes

characteristic

C₂×Dic₁₂₂

—

C₂×C₈×C₆₁

Q₈

yes

complemented

C₂×C₈×C₆₁

C₂³

yes

—

C₂×C₈×C₆₁

C₂³

yes

—

C₄²×C₆₁

D₄

yes

—

C₄²×C₆₁

C₂³

yes

characteristic

C₄²×C₆₁

C₂³

yes

characteristic

C₂²⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

C₂³

yes

—

#11758196

—

C₄×Dic₆₁

—

C₂³×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂×D₄

yes

—

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

C₂×D₄

yes

—

C₂×C₄×C₆₁

C₂×Q₈

yes

—

C₂×C₄×C₆₁

C₂⁴

yes

characteristic

C₂×Dic₆₁

—

Dic₁₂₂

—

C₈×C₆₁

C₂×Q₈

yes

—

C₈×C₆₁

C₂×Q₈

yes

—

C₂²×C₆₁

—

C₂²×C₆₁

C₄⋊Q₈

yes

—

C₂²×C₆₁

C₂²×D₄

yes

characteristic

C₄×C₆₁

—

C₄×C₆₁

C₄⋊Q₈

yes

—

C₄×C₆₁

#135

yes

—

C₄×C₆₁

C₂²×Q₈

yes

—

Dic₆₁

—

#3160

—

maximal, Sylow

C₂×C₆₁

—

C₂×C₆₁

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₆₁

C₂×#135

yes

—

—

—

—

—

—

C₂×C₄×C₈

D₆₁

yes

characteristic

C₆₁

#3160

yes

characteristic, complemented, Sylow

C₂²×C₈

C₂×D₆₁

yes

—

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

122

—

C₄×Q₈

—

122

—

#117

—

C₂×C₄²

C₂×D₆₁

yes

characteristic

#107

—

#106

—

C₄×C₈

C₂×D₆₁

yes

characteristic

C₄×C₈

C₂×D₆₁

yes

—

C₄×C₈

C₂×D₆₁

yes

characteristic

C₂×Q₈

—

122

—

C₂²×C₄

D₂₄₄

yes

—

C₂²×C₄

C₂²×D₆₁

yes

characteristic

C₂×C₈

Dic₁₂₂

yes

complemented

C₂×C₈

C₂²×D₆₁

yes

—

C₂×C₈

C₂²×D₆₁

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

122

—

C₄⋊C₄

—

122

—

C₂²⋊C₄

—

122

—

C₄²

—

122

—

C₄²

D₂₄₄

yes

—

C₄²

C₂²×D₆₁

yes

characteristic

C₄²

C₂²×D₆₁

yes

characteristic

C₂³

C₂×D₂₄₄

yes

characteristic

Q₈

—

244

—

C₂×C₄

—

C₂×C₄

—

122

—

C₂×C₄

C₂×Dic₁₂₂

yes

—

C₂×C₄

C₂×D₂₄₄

yes

—

C₂×C₄

C₂×D₂₄₄

yes

—

C₂×C₄

C₂×D₂₄₄

yes

characteristic

C₂×C₄

C₂³×D₆₁

yes

characteristic

C₈

C₂×Dic₁₂₂

yes

—

C₈

C₂×Dic₁₂₂

yes

—

100

C₂²

—

101

C₂²

#14519777

yes

—

102

C₂²

C₂²×D₂₄₄

yes

characteristic

103

C₄

—

104

C₄

—

244

—

105

C₄

#14519777

yes

—

106

C₄

#14519799

yes

—

107

C₄

C₂²×Dic₁₂₂

yes

—

108

C₂

—

109

C₂

C₂×#14519777

yes

characteristic

110

C₂

C₂×#14519799

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×D₂₄₄
commutator	C₂×C₄×C₆₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₆₁
Fitting	C₂×C₄×C₈×C₆₁	C₂
socle	C₂²×C₆₁	C₂²×D₄
hypercenter	C₂×C₄×C₈	D₆₁
hypocenter	C₆₁	#3160
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂×C₄

C₂×D₂₄₄

commutator

C₂×C₄×C₆₁

C₂⁴

Frattini

C₂×C₄

C₂³×D₆₁

Fitting

C₂×C₄×C₈×C₆₁

C₂

socle

C₂²×C₆₁

C₂²×D₄

hypercenter

C₂×C₄×C₈

D₆₁

hypocenter

C₆₁

#3160

LCS

G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

N	H ≅ G/N	type
2. #16840167	C₂	semidirect
3. #16840164	C₂	semidirect
4. #16840160	C₂	semidirect
22. C₂×C₈×C₆₁	Q₈	semidirect
62. C₆₁	#3160	semidirect
78. C₂×C₈	Dic₁₂₂	semidirect

H ≅ G/N

type

2. #16840167

C₂

semidirect

3. #16840164

C₂

semidirect

4. #16840160

C₂

semidirect

22. C₂×C₈×C₆₁

Q₈

semidirect

62. C₆₁

#3160

semidirect

78. C₂×C₈

Dic₁₂₂

semidirect

Subgroups — #21456333

Conjugacy classes of subgroups

Complements